Balanced dualizing dg-module for every Gorenstein dg-algebra

Prove that for any Gorenstein differential bigraded k-algebra A, if integers a and n satisfy R_A(k, A) ≅ k(a)[−n] in the derived category D(A), then the dg-A–A-bimodule R = A(−a)[n] is a balanced dualizing dg-module; that is, R is a dualizing dg-module and moreover the derived m-torsion satisfies RΓ_m(R) ≅ Hom_k(A, k) ≅ RΓ_{m^{op}}(R) in D(A^{op} ⊗_k A), where m and m^{op} are the homogeneous maximal ideals of H^0(A) and H^0(A^{op}), respectively.

Background

The paper introduces a dg-analogue of Yekutieli’s balanced dualizing complex for graded algebras and proves Serre duality for noncommutative spaces associated to dg-algebras under the existence of a balanced dualizing dg-module. A central technical result (Theorem 3.) establishes the existence of such a module when the dg-algebra is graded commutative and Gorenstein; a finite-dimensional cohomology case is also covered (Proposition 3.).

Removing graded commutativity would extend Yekutieli’s existence results for noncommutative Gorenstein k-algebras to the dg setting and would allow several key theorems in the paper (e.g., finiteness of derived global sections and the dg version of Orlov’s correspondence) to hold for general Gorenstein dg-algebras without extra assumptions.

As partial evidence, the authors note that for R = A(−a)[n] one can identify RΓ_m(R) and Hom_k(A, k) as complexes of k-vector spaces; the remaining difficulty is to produce a quasi-isomorphism of A–A-bimodules. The obstacle is the lack of a suitable theory of minimal injective resolutions for noncommutative dg-algebras, which is crucial in the graded commutative case via Minamoto’s results.